SEMI-ANALYTIC CALCULATION OF

THE SHIFT IN THE CRITICAL

TEMPERATURE FOR BOSE-EINSTEIN
CONDENSATION

Dr. Eugeniu Radescu

OUTLINE

General Notions
The problem of Tc
Method
Results
Conclusions
2

Introduction
According to Louis de Broglie, very cold particles behave like waves
whose wavelengths increase as their velocity drops. The particle is
delocalized over a distance corresponding to the de Broglie wavelength
dB = h/mv,
where h = h/2 is Plancks constant.
When a gas is cooled down to very low temperatures, the individual
atomic de Broglie waves become very long and eventually overlap.
If the gas consists of bosonic particles all being in the same quantum state, the de Broglie waves of the individual particles constructively interfere and build up a large coherent matter-wave.

 The transition from a gas of individual atoms to the macroscopic

quantum state occurs as a phase transition and is named BoseEinstein condensation (BEC) after Shandrasekhar Bose and Albert
Einstein.
Bose-Einstein condensation occurs as a phase transition from a gas
to a new state of matter that is in many respects dissimilar to all
usual macroscopical states such as solids, liquids or gases.
The experimental verification of Bose-Einstein condensation has
been achieved in 1995 (Anderson et al., Davis et al., Bradley et
al.). The observation of Bose-Einstein condensation has now been
confirmed by more than twenty groups worldwide and triggered an
enormous amount of theoretical and experimental work on Bosecondensed gases.

Basic Notions
For a gas composed of particles of mass m at temperature T , the
velocity distribution is given by the Maxwell-Boltzmann law:
g(v) =

2kB T

mv2
exp
2kB T

where kB is the Boltzmann constant.

This law describes well the behavior of atoms at low density and
high temperatures. Deviations from it are insignificant until quantum mechanical effects assert themselves, and this does not occur
until the temperature becomes so low that the atomic de Broglie
wavelength dB becomes comparable to the mean distance between
particles.

 Since the typical momentum mv is of the order (mkB T )1/2 , the

de Broglie wavelength of a typical particle is of order the thermal
wavelength defined by
2h2
,
mkB T

T =

For a general system with density n, the mean distance between

particles is n1/3 . Quantum effects are expected to show up for
n1/3 T .
Example: an atomic gas at room temperature (T 300K) and with
the density of air at sea level (n 3 1019 cm3 ) is safely within the
T = 1 1010 cm.
classical regime, since n1/3 3 107 cm
To witness quantum effects, one needs atoms at low temperature
and relatively high density.

Bose-Einstein Condensation of Ideal Gas

of Bosons

Ideal gas is a gas of point-like noninteracting particles.

Bosons follow a quantum statistical distribution called Bose-Einstein
distribution. The basic difference between Maxwell-Boltzmann
statistics and Bose-Einstein statistics is that the former applies to
particles with the same mass that nevertheless are distinguishable
from one another, while the latter describes identical indistinguishable particles.
Bosons, in contrast to fermions, enjoy sharing a quantum state and
even encourage other bosons to join them.

Bose-Einstein distribution for a non-degenerate quantum state with

energy when the system is held at temperature T is:
f () =

1
e() 1

where 1/kB T and 0 is chemical potential.

The expression for the number density of particles is
n

2 2m
h3

1/2 d
.
e() 1

Since 0, this expression suggests that for fixed T , there is a

maximum number density n = N/V :

2 2m
n
h3

1/2 d
= (3/2)

e 1

m
2h2

3/2

(kB T )3/2 .

 Equivalently, for fixed number density, there is a minimum temperature Tc to which the bosons can be cooled.
2 h2 n2/3
.
kB Tc = 3 2/3
m
( 2 )
Einstein pointed out that there is no minimum temperature. Instead, for T < Tc there is a macroscopic occupancy of the ground
state. The number N0 of atoms in the = 0 state is
N0 = N 1

T
Tc

3/2

Atomic Interactions in Nonideal

Homogeneous Bose-Gas
Real systems are always affected by particle interactions.
The formula for Tc implies that the interparticle spacing n1/3 and
the thermal wavelength T = (2h2 /mkB T )1/2 are comparable.
We assume that they are both large compared to the range R of
R.
the two-body potential: n1/3 , T
In this case, the interaction between the bosons can be characterized entirely by the s-wave scattering length a.
We assume a to have magnitude of order R. Thus we require
a

n1/3 , T .

10

Shift in Tc
If a is the only parameter, then an1/3 is dimensionless and the
transition temperature must take the form
Tc (a) = Tc (a = 0) f (an1/3 )
The leading order shift in Tc is dominated by the infrared region
of momentum space (the typical momenta involved are of order
1/T ) and is insensitive to the ultraviolet region (Baym,
a/2T
1999)
Effective field theory reveals that as an1/3 0, the leading correction is linear in an1/3 :
Tc (a) = Tc (a = 0){1 + c a n1/3 + O(a2 n2/3 )},
where c is a numerical constant.
11

 Several theoretical predictions for the value of the coefficient c can

be found in literature. The values span a range from 1 to 5.
Recently (Kashurnikov et al. and Arnold et al., 2001), the lattice
Monte Carlo simulations gave a definitive answer to the problem
of finding the leading order correction to the shift in Tc , giving for
the coefficient c the value
Tc
= c an1/3 ,
Tc
c = 1.32 0.02
Is there a simpler way to derive this quantity?

12

Wilkens et al. [2]

0
Gruter et al. [3]
Holzmann et al. [4]

1
2

Figure 1:
13

Arnold, Tomasik (NLO 1/N) [5]

Holzmann, Krauth [6]
Baym et al. (LO 1/N) [7]

Baym et al. [1]

de Souza Cruz et al. [8]

4
Stoof [9]

Method
This system can be described by a second-quantized Schrodinger
equation. The corresponding imaginary-time Lagrangian is
L =

1 2
2a 2

+
( ) ,

2m
m

For simplicity we use units such that h = kB = 1.

The field (x, t) can be decomposed into Fourier modes n (x) with
frequencies n = 2nT. At sufficiently large distance scales ( T )
and small chemical potential (||
T ), the n = 0 frequency modes
decouple from the dynamics leaving an effective theory of only the
zero modes (Baym, 1999).

14

 The effective lagrangian density for the dimensionally reduced theory can then be written as
Leff

1
1
1 2
2
= + r + u 2
2
2
24

where we replace the complex field 0 by the 2-component real

field = (1 , 2 ) defined by
0 (x) = (mT )1/2 (1 (x) + i2 (x)) ,
and the new parameters are
r

= 2m,

u = 48amT.

15

 The leading order shift Tc in the critical temperature at fixed

number density is related to the leading order shift nc in the
critical density at fixed temperature by
2 nc
2 mT
Tc
=
=
,
Tc
3 nc
3 nc
where has a diagramatic expansion within the framework of the
effective theory.
The number density to the accuracy required to calculate the shift
in Tc to leading order in an1/3 , is

n = n0 + mT 2 ,
where n0 is the short-distance contribution (coming from momentum scales 1/T ).

16

Figure 2: Diagram for 2 at the critical point. The black blob represents the 2 vertex, while the shaded blob represents the complete
propagator with subtracted self-energy (p) (0).

17

 2 can be expressed as
2 = 2

p2 + r + (p)

where (p) is the self-energy of the field.

can be calculated within the effective 3-dimensional field theory:
2

=2
p

p + (p) (0)

2 1

We made use of the condition that the correlation length at the

critical point is infinite:
r + (0) = 0.
At the phase transition, there is only one relevant length scale and
it is set by the parameter u a. Since has the same dimension
as u, it must be proportional to u by simple dimensional analysis,
and therefore Tc is linear in a.
18

 Although has a well-defined diagramatic expansion, any attempt

to calculate Tc using ordinary perturbation theory in u is doomed
to failure.
Nonperturbative methods that have been applied to this problem include numerical simulations, large-N techniques, variational
methods and self-consistent schemes.

19

Linear -expansion

The linear -expansion (LDE) is a particularly simple variational

method for obtaining nonperturbative results using perturbative techniques. The convergence properties of the LDE have been studied extensively for the quantum mechanics problem of the anharmonic oscillator.
It can be defined by a lagrangian whose coefficients are linear in a
formal expansion parameter .
If L is the lagrangian for the system of interest, the lagrangian that
generates the LDE has the form
L = (1 )L0 + L,
where L0 is the lagrangian for an exactly solvable theory.

20

 In our case the lagrangian L can be written L = L0 + Lint , where

1
1
L0 = 2 + m 2 2 ,
2
2
2

u
2
2
2
rm +

.
Lint =
2
24
Calculations are carried out by using as a formal expansion parameter, expanding to a given order in , and then setting = 1.
The lagrangian L0 for the exactly solvable theory involves an arbitrary dummy parameter m which is treated as a variational parameter.
A prescription for m is required to obtain a definite prediction.
A simple prescription for m is the principle of minimal sensitivity
(PMS) that the derivative with respect to m should vanish (Stevenson,1981).
The LDE has been proven to converge in the Anharmonic Oscillator calculations for appropriate order-dependent choices of the
variable m that include the PMS criterion as a special case.
21

 A previous application of the LDE to our problem gave a result

which scales incorrectly with the number of the scalar fields in the
theory (de Souza Cruz et al., 2001).
To circumvent this problem, we apply LDE only to the self-energy
part (p) (0) of the propagator.

22

Large N limit

The large-N limit is defined by N , u 0 with N u fixed, where

N is the number of scalar fields in the Lagrangian (we set N = 2 at the
end of the computation, making the contact with the initial Lagrangian.)

Figure 3: The 4th in the series of diagrams for (p) that survive in the
large-N limit.
In the large-N limit, the shift in Tc can be computed analytically.
23

 The series in in the large-N limit involves a particular class of

self-energy diagrams which can be computed numerically to a very
high order in expansion. In this way we are able to see the rate of
the convergence to the analytic result.
It is easy to show that if the LDE converges, it converges to the
correct analytic result.
The LDE seems to converge for all above some critical value near
0.7.
The PMS prescription improves the convergence rate by allowing
the variational parameter to vary with the order in .

24

1.5

/(-Nu/96 )

n=4
1

n=35
n=5
n=3

0.5

0.6

0.8

25

1.2

1.4

n
3
4
5
7
11
19
35

0.481
0.611
0.675
0.742
0.801
0.850

Table 1: The values of /(N u/96 2 ) in the large-N limit at nth order
in the . The analytic result is equal to 1.

26

Full O(N ) (N = 2) effective theory

After the results (convergence, but a very slow convergence) for the
particular class of large-N diagrams, we are ready to compute several
orders of the full O(N ) theory. The 1PI self-energy diagrams needed are
shown in the following figures.

27

Figure 5: The diagrams that contribute to (p) (0) at order 3 .

Figure 6: Four-loop diagrams that contribute to (p) (0) at order 4 .

28

29

 For N = 2, the 3rd , 4th and 5th order approximations differ from
the lattice Monte Carlo result (0.57) by about 66%, 63% and 59%
respectively. These predictions are not as accurate as in the large N
limit, where the errors in the 3rd , 4th and 5th order approximations
are about 52%, 44% and 40% respectively.
The LDE seems to be approaching the correct result for N = 2,
albeit very slowly.

30

n
2

/(u/48 2 )

0.192

0.2088

0.2373

Table 2: The values of /(u/48 2 ) for N = 2 at nth order in the LDE.

The lattice result is 0.57 0.1

31

10

Variational Perturbation Theory (VPT)

applied to the problem of the shift in Tc

The approach to a second order phase transition is characterized

by nontrivial critical exponents that differ from the values predicted by
dimensional analysis. For example, the scaling dimension of the scalar
field changes from the naive value d/2 1 (1/2 in our case) to d/2
1 + /2, where 0.033 is a critical exponent.
This information can be incorporated into a variational procedure
which accelerates the convergence compared to the application of
the LDE.
The method introduces a new variational parameter q which governs the behavior of the quantity in the strong coupling (massless) limit.

32

 The method uses the fact that the quantity /u approach a constant as the weak-coupling expansion parameter u/m goes to infinity.
/u = f (u/m) f ().
(1)
We will use the PMS prescription to fix the parameters s and q at
each order in the expansion.
For q = 1 the VPT method reduces to the LDE method.
VPT method increases the rate of the convergence, by having q as
a variational parameter allowing m to approach 0 with a nontrivial
exponent.
We know only the truncated weak-coupling expansion
N

an (u/m)n

fn (u/m) =
n=1

Construct function Fn (u/m, q) by

a) setting u u, m m(1 )q
33

(2)

b) truncate after nth order in

c) setting = 1
N

An (q)(u/m)n

Fn (u/m, q) =

(3)

n=1

Modulo interchange of limits

Fn (u/m, q) f ().

34

(4)

-0.5

<Fi^2>

-0.6
-0.7
-0.8
-0.9
-1
4

8
n

10

12

Figure 8: /(N u/96 2 ) in the large-N limit for n from 4 to 12. q = 1

(LDE) results are shown with empty triangles. Empty squares are for
fixed value q = 2. Results for variationally determined values of qn are
shown with black squares. The analytic result 1 is shown with dashed
line.
35

0
-0.1

<Fi^2>

-0.2
-0.3
-0.4
-0.5
-0.6
4

Figure 9: /(N u/96 2 ) for the O(2) theory for F2opt , F3opt , F4opt . q = 1
(LDE) results are shown with empty triangles. Empty squares are for
fixed value q = 2. Results for variationally determined values of qn are
shown with black squares. The lattice result is shown with two lines
denoting the maximal and the minimal value separated by 2 error bars.
36

11

Conclusions

This work is among the few applications of the LDE to a nonperturbative field-theoretical problem which gives unequivocal results.
It shows the importance of the correct understanding of the energy
scales involved in the problem. The power of an1/3 , the sign of
the coefficient and even its order of magnitude can be deduced
without calculation just by applying correctly the effective field
theory method.
The LDE shows itself as a systematically improvable scheme to
compute quantitatively nonperturbative quantity. Applied to the
problem of Tc , it seems to converge very slowly towards the correct
result.
By incorporating the knowledge about the behavior of the strongcoupling limit, VPT shows itself as a strong tool which competes
very well with lattice Monte Carlo simulations.

37